This paper produces a follow-up to a recent Journal of Algebra paper by M.D.E. Conder and D.W. Young, extending the main theorem of the latter paper to arbitrary co-compact Fuchsian groups. In particular, it shows that every non-perfect co-compact Fuchsian group has a smooth finite soluble quotient of derived length c ≤ 3 and classifies those for which c = 1 or c=3 . In this way it also extends and generalizes certain theorem by Chih-Han Sah published in Acta Mathematica of 1969 and improves main theorem of R. Zomorrodian published in Illinois J. Math. of 2007.
We introduce a new class of affine connections on Hessian manifolds, called hybrid connections, characterized by the compatibility between their projective geometry with the underlying affine structure on the one hand and their infinitesimal holonomy with the Hessian metric on the other hand. In this paper, we investigate the properties of hybrid connections and prove that, on a given Hessian manifold, they are completely determined by the choice of a Hessian potential for the metric. In the special case of pseudo-Euclidean manifolds, we identify canonical models and construct, in particular, a natural connection on the open unit ball that combines features of the Cayley–Klein and Poincaré models of hyperbolic geometry. We also prove the existence and uniqueness (up to scaling) of a pseudo-Riemannian metric h such that the geodesics of ∇ admit parameterizations of constant speed with respect to h, which we call the isochrone metric.
In this paper, we study n-dimensional (n≥ 3) conformally flat minimal Lagrangian submanifolds in complex space forms. Our main result is a classification of such submanifolds under the assumption that the Ricci tensor is semi-parallel. This generalizes the recent classification result of Song–Wang–Xing [1] on conformally flat minimal Lagrangian submanifolds in complex space forms with parallel Ricci tensor.
Let S⊂ℝ^3 be a smooth embedded sphere whose normal curvatures have absolute value at most 1. We prove that if S is contained in an open ball of radius 2, then the body bounded by S contains a unit ball.
It is well-known that the Euclidean plane has a standard 6-regular triangulation by equilateral triangles and the hyperbolic plane has a k-regular geometric triangulation for every k>6 . In this note, we construct 6-regular geometric triangulations of the hyperbolic plane with a uniform bound on edge lengths, resolving a conjecture by Feng Luo [6]. We also show that there is no uniform bound on stretch factors of Delaunay triangulations in the hyperbolic plane, in contrast to the existence of such a bound in the Euclidean plane.
